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El cateto más corto de un triángulo rectángulo es 7 cm más corto que el cateto más largo. La hipotenusa es 7 cm más larga que el
cateto más largo. Hallar las longitudes de los lados del triángulo.
Longitud del cateto más corto:
Longitud del cateto más largo:

Ask by Bowers West. in Mexico
Mar 29,2025

Solución de inteligencia artificial de Upstudy

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El cateto más corto mide 21 cm, el cateto más largo mide 28 cm, y la hipotenusa mide 35 cm.

Solución

  1. Sea la longitud del cateto más largo. Entonces, el cateto más corto es y la hipotenusa es .
  2. Según el teorema de Pitágoras:
  3. Desarrollamos los cuadrados:
  4. Sustituyendo en la ecuación:
  5. Combinamos términos semejantes:
  6. Restamos en ambos lados de la ecuación:
  7. Factorizamos:
  8. Descartamos la solución , ya que las longitudes deben ser positivas. Por lo tanto, cm.
  9. Entonces:
    • Longitud del cateto más largo: cm.
    • Longitud del cateto más corto: cm.
    • Longitud de la hipotenusa: cm.

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Bonus Knowledge

Para resolver este problema, primero llamemos al cateto más corto “a” y al cateto más largo “b”. Según la información, tenemos que:
  1. (el cateto más corto es 7 cm más corto)
  2. La hipotenusa “c” es .
Ahora, aplicamos el teorema de Pitágoras:
Sustituyendo a y c:
Expandimos y simplificamos:
Restamos de ambos lados:
Factorizando, obtenemos:
Por lo tanto, cm (el cateto más largo) o (no válido). Así que, el cateto más corto será:
Por lo tanto, las longitudes de los lados del triángulo son:
Longitud del cateto más corto: 21 cm
Longitud del cateto más largo: 28 cm.

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